The Rewards of the Infinite

September so soon.

With any luck, the approach of May will be just as swift. Still, one has doubts. On the other hand, what with the apocalyptic combinations of El Niño, Global Warming, and the End of the Millenium, perhaps next frame is too distant by far. We beg you to take a moment to read about what may be the biggest environmental issue of all time

Alchemy is never far from the mind of Paracelsus, and this week is no exception.
Q: What’s the parlor game that changed the world?
A: Ask Zeno. You’re halfway there already.

The Rewards of the Infinite

Alchemists spent their time in pursuit of something called the Philosopher’s Stone (also known as the Lapis Philosophorum or the Sophic Hydrolith). This stone, once manufactured, acted as a sort of catalyst in a number of procedures, of which the most lucrative was the transmutation of base metals such as lead or iron into gold. The stone, if it could be tamed, thus acted as a kind of gateway to a realm of limitless wealth. Not surprisingly, it was considered extremely troublesome to create and difficult to handle. Because it was more often discussed than seen, it also became the subject of much philosophical speculation. In alchemical literature a variety of symbols were assigned to it, including

or more simply

These symbols were considered powerful in and of themselves and as such were closely guarded secrets.

A hasty reading of alchemy will always reveal the labor of fools; no Philosopher’s Stone ever came to light, only a tantalizing collection of anecdotes. There are, however, remarkable parallels between alchemy and more reputable Western studies. The link between alchemy and chemistry is obvious and direct, but I would go so far as to say that in mathematics, we have truly found our Philosopher’s Stone in the taming of the infinite.

To the ancient Greeks, the word for infinity, apeiron, was pejorative. It was used as a kind of cudgel on certain problems, most famously by Zeno of Elea in his famous paradoxes. When the infinite entered a problem, all bets were off, and mathematicians simply had to walk away. Consider Zeno’s paradox of the race between Achilles and the Tortoise. Achilles can run ten times faster than the Tortoise, but the Tortoise has a 100 yard head start. In the time it takes Achilles to run 100 yards, the Tortoise will forge ahead ten. Things look bad for the Tortoise, but remember that in the time it takes Achilles to run the next ten yards, the Tortoise will gain another yard. If we continue our analysis along these lines, it becomes evident that Achilles can never catch the Tortoise, despite the fact that if the race were actually run, we would see that after precisely 111 1/9 yards, Achilles will surge into the lead. Why did the math let us down? The short answer is that the Greeks had no formulation, no symbol, for infinity. An encounter infinite simply meant that no further progress could be made on that problem.

In the medieval period, when the alchemists were vainly pursuing their Stone, mathematicians were still being vexed by the infinite. Hot debates centered around whether a series like

1 – 1 + 1 – 1 + 1 – 1 + …

converged to a meaningful sum or not. Because such disputes could not be resolved, problems of the infinite reached a chicken-and-egg sense of futility. The main point here is that, despite the promise of usefulness, any attempt to manage the infinite in a mathematical formulation was considered doomed and foolish from the outset, much as we now view the promise of the Lapis Philosophorum.

The tide finally began to change with the arrival of Leibniz and Newton. Both men conceived the idea of infinitesimals, which is to say, things so small that they have no significance unless there are an infinite number of them. The invention of calculus by these men brought to fruition fantastically useful calculations that had completely baffled earlier generations of mathematicians and scientists.

Consider the case of a falling stone, whose distance fallen is given by the formula

s = 16 t²

where s is the distance in feet and t is the time in seconds. What is the velocity of the stone at time t = 1 seconds? If dt stands for the infinitesimal increment of time and ds stands for the infinitesimal increment of distance, then the instantaneous velocity we’re after is the ratio ds/dt. By considering the distance traveled by the stone between t = 1 (when the rock has gone 16 feet) and t = 1 + dt (when the rock has gone 16 x (1 + dt)² feet), we can work out the answer. Don’t let the math slow you down too much; the answer comes out to be ds/dt = 32 + 16 dt feet/second. In a practical sense, this is good enough. Just pretend like dt = 0 and you’ve computed a perfectly good answer: 32 feet/second.

But on second thought, what the hell is that dt thing? Is it really zero, or not? What’s really fascinating here is that even Leibniz and Newton couldn’t really explain this. Because the rest of the math was turning out okay, they effectively shrugged their shoulders and moved on. Not everyone was content to let them get away with it; Bishop George Berkeley thought it was tantamount to a statement of religious faith, therefore deserving no place in mathematical theory. He summed up his views in 1734 with a brutal critique entitled (in characteristic 18th century loquaciousness) The Analyst, Or A Discourse Addressed to an Infidel Mathematician. Wherein It is examined whether the Object, Principles, and Inferences of the modern Analysis are more distinctly conceived, or more evidently deduced, than Religious Mysteries and Points of Faith. “First cast out the beam out of thine own Eye; and then shalt thou see clearly to cast out the mote out of thy brother’s Eye.” This is the last serious use of the Greek cudgel against calculus, but matters of the infinite still stir up controversy.

Eventually infinitesimals as such were replaced by the use of limits, but it is a little known fact that it wasn’t until the time of the German mathematician Kurt Weierstrass in the late 19th century that the book was finally closed on Bishop Berkeley’s objections. By then, of course, the promise of calculus was already delivering enormous results. Consider the impact of calculus on our society: from spaceflight to architecture to econometrics, differential equations are simply everywhere. It is an impossible exercise to imagine modern life without them.

Finally, look at this equation: 1/2 + 1/4 + 1/8 + 1/16 + … = 1. How can we be so certain of the result on the right hand side? Because we have come to grips with the infinite. We’ve embodied it in its symbol in the following restatement of the above equation.

The hopeless striving of the dot dot dot in the series is replaced by a compact symbol, and a gateway to limitless wealth is opened up. As with many powerful concepts, we cannot claim to truly understand it, but at least we can put it to work.